Let $\alpha $ be a complex valued $2$-cocycle of finite order of a finite group $G.$ The nth Frobenius–Schur indicator of an irreducible $\alpha $-character of G is defined and its properties are investigated. The indicator is interpreted in general for $n =2$ and it is shown that it can be used to determine whether an irreducible $\alpha $-character is real-valued under the assumption that the order of $\alpha $ and its cohomology class are both $2$. A formula, involving the real $\alpha $-regular conjugacy classes of $G,$ is found to count the number of real-valued irreducible $\alpha $-characters of G under the additional assumption that these characters are class functions.

Let $\alpha $ be a complex-valued $2$-cocycle of a finite group G with $\alpha $ chosen so that the $\alpha $-characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $\alpha $-regular are characterised by the values that the irreducible $\alpha $-characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of $G;$ however, the initial choice of $\alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.

In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$-groups with generalized corank at most three.

Let H be a subgroup of a finite group G and let $\alpha $ be a complex-valued $2$-cocycle of $G.$ Conditions are found to ensure there exists a nontrivial element of H that is $\alpha $-regular in $G.$ However, a new result is established allowing a prime by prime analysis of the Sylow subgroups of $C_G(x)$ to determine the $\alpha $-regularity of a given $x\in G.$ In particular, this result implies that every $\alpha _H$-regular element of a normal Hall subgroup H is $\alpha $-regular in $G.$

Let $\alpha $ be a complex-valued $2$-cocycle of a finite group $G.$ A new concept of strict $\alpha $-regularity is introduced and its basic properties are investigated. To illustrate the potential use of this concept, a new proof is offered to show that the number of orbits of G under its action on the set of complex-valued irreducible $\alpha _N$-characters of N equals the number of $\alpha $-regular conjugacy classes of G contained in $N,$ where N is a normal subgroup of $G.$

The purpose of this paper is a further investigation on the 2-nilpotent multiplier, $\mathcal{M}$(2)(G), when G is a non-abelian p-group. Furthermore, taking G in the class of extra-special p-groups, we will get the explicit structure of $\mathcal{M}$(2)(G) and will classify 2-capable groups in that class.

We use Godeaux–Serre varieties of finite groups, projective representation theory, the twisted Atiyah–Segal completion theorem, and our previous work on the topological period-index problem to compute the étale index of Brauer classes α ∈ Brét(X) in some specific examples. In particular, these computations show that the étale index of α differs from the period of α in general. As an application, we compute the index of unramified classes in the function fields of high-dimensional Godeaux–Serre varieties in terms of projective representation theory.

Let $F$ be an algebraically closed field of characteristic $p$, and let $H$ be a finite group. A natural and important problem in representation theory is to classify the pairs $(G, D)$, where $G$ is a subgroup of $H$, and $D$ is an irreducible $FH$-module that remains irreducible when restricted to $G$. For example, if $H$ is an almost simple group or a central extension of an almost simple group, then one encounters this problem in the attempt to determine the maximal subgroups of the finite classical groups. We solve the above problem for Schur's double covers $H = {\widehat S}_n$ and ${\widehat A}_n$ of the symmetric and alternating groups, and $G$ a Young-type subgroup. The answer is given in terms of the combinatorics of restricted, $p$-strict partitions; such partitions are used to parameterize the irreducible spin representations of ${\widehat S}_n$ and ${\widehat A}_n$. We exploit the connections, recently developed by Ariki, Grojnowski, Brundan, Kleshchev, and others, between modular branching rules, crystal graphs of affine Kac–Moody algebras, and affine Hecke algebras and related objects.

In this note we further our investigation of Baer invariants of groups by obtaining, as consequences of an exact sequence of A. S.-T. Lue, some numerical inequalities for their orders, exponents, and generating sets. An interesting group theoretic corollary is an explicit bound for $|{{\gamma }_{c+1}}\,(G)|$ given that $G\,/\,{{Z}_{c}}\,(G)$ is a finite $p$-group with prescribed order and number of generators.

We provide an isomorphism between the Grothendieck ring of modules of the twisted quantum double of a finite group, and a product of centres of twisted group algebras of centralizer subgroups. It follows that this Grothendieck ring is semisimple. Another consequence is a formula for the characters of this ring in terms of representations of twisted group algebras of centralizer subgroups.

Let U(KλG) be the group of units of the infinite twisted group algebra KλG over a field K. We describe the FC-centre ΔU of U(KλG) and give a characterization of the groups G and fields K for which U(KλG) = ΔU. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus theorem.

Three main results are obtained in this paper: one generalizes the Atiyah-Bott-Shapiro periodicity equivalence on the category of real Clifford modules, (Theorem 2.2); another establishes the existence of two algebras for real projective representations of the symmetric group Sn and the alternating group An, (Theorem 3.2) and determines their structure, (Theorem 6.1); the third describes all the real projective representations of Sn and An except for some small numbers n, (Theorem 7.2).

Motivated by the problem of giving a functorial (or at least uniform) description of the projective representations for wreath products Gʅ Snin terms of those for G, we study a certain binary operation on the class of “cyclic covering groups with parities”. Along with setting up the basic machinery associated to representations graded by (Ζ/2)ℓ , the main result is a description of the irreducibles for in terms of a (tensorlike) product of those for Aand for B.Finally we describe a programme for producing a PSH-algebra theory in this context, analogous to that of Zelevinsky for the case ℓ=0, and that of the author with with Michael Bean (structure) and with John Humphreys (applications) for the case ℓ=1